# l18 - CIS 540 Principles of Embedded Computation Spring...

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CIS 540 Principles of Embedded Computation Spring 2016 Instructor: Rajeev Alur [email protected]

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Dynamical Systems Controller interacting with the physical world via sensors and actuators Thermostat controlling temperature Cruise controller regulating speed of a car Variables: Physical quantities evolving continuously over time Temperature, pressure, velocity … Continuous-time models using differential equations CIS 540 Spring 2016; Lecture March 28
Model-Based Design Block Diagrams Widely used in industrial design Tools: Simulink, Modelica, RationalRose… Key question: what is the execution semantics? Similar to synchronous model, but continuous-time instead of discrete-time CIS 540 Spring 2016; Lecture March 28

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Traditional Feedback Control Loop Plant Controller Actuators Sensors CIS 540 Spring 2016; Lecture March 28
Example: Heat Flow Input variables: h in (heat inflow) and h out (heat outflow) of type real h net = h in - h out Output variable: h net (net heatflow) of type real No state variables Signal: assignment of values to variables as function of time t At each time t, value of output signal h net (t) equals h in (t) – h out (t) Output as a function of inputs/state specified using algebraic equations (as opposed to assignments) h net h in h out CIS 540 Spring 2016; Lecture March 28

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Car Model Position x Velocity v Force F Friction k v Newton’s law of motion gives F – kv = m d 2 x/dt CIS 540 Spring 2016; Lecture March 28
Continuous-time Component Car v F dx = v; dv = (F - kv) / m; real x L <= x <= x U ; v L <= v <= v U For each state variable x, its rate of change dx/dt defined using an expression over input and state variables For each output variable, its value is defined using an expression over input and state variables. CIS 540 Spring 2016; Lecture March 28

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Executions of Car Input signal: Function F (t) : real >=0 -> real that gives value of Force as a function of time Should be continuous or piecewise-continuous Given an initial state (x 0 , v 0 ) and input signal F (t), the execution of the system is defined by state-signals x (t) and v (t) (from real >=0 to real) that satisfy the initial-value problem: x (0) = x 0 ; v (0) = v 0 d x (t)/dt = v (t) d v (t)/dt = ( F (t) – k v (t)) / m CIS 540 Spring 2016; Lecture March 28
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• Fall '09
• ALUR
• Lectures, Continuous function, Dynamical systems, Lipschitz continuity

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